Finite Group Databasebeta

← #13154267

Subgroups — #13154267

Order 1728.

1716 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88744C212yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#11721850C212yesmaximal, complemented
4#11721849C214yesmaximal, complemented
5#11721848C212yesmaximal, complemented
6D4×#1084C212yesmaximal, complemented
7C4×C4⋊#272C211yesmaximal, characteristic, complemented
8C4×#6229C211yesmaximal, characteristic, complemented
9C3×#5718—34nomaximal
10C2×D4×H(3,𝔽3)C2212yes—
11C2×D4×H(3,𝔽3)C2212yes—
12C2×D4×H(3,𝔽3)C2212yes—
13C2×#6235C2214yes—
14C2×#6235C2214yes—
15C22×#1084C2214yes—
16#88744—28no—
17C2×C4⋊#272C2211yescharacteristic
18C2×C4×#272C2212yes—
19C2×#6229C2211yescharacteristic
20C42×H(3,𝔽3)C2211yescharacteristic
21#88552C2214yes—
22#88542C2212yes—
23C4⋊#1084C2211yescharacteristic
24C4⋊#1084C2211yescharacteristic
25#88540C2212yes—
26C4×#1084C2212yes—
27C2×C3×#237—38no—
28C32×#127D314yescomplemented
29C3×#940—38no—
30C3×#939—316no—
31C3×#938—38no—
32C3×D4×Dic3—38no—
33C3×C4×D12—34no—
34C3×C4×Dic6—34no—
35C23×H(3,𝔽3)C2314yes—
36C22×#272C2312yes—
37D4×H(3,𝔽3)—24no—
38D4×H(3,𝔽3)—24no—
39D4×H(3,𝔽3)—42no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—28no—
44#6235—44no—
45C2×#1084—28no—
46C2×#1084C2312yes—
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49C4⋊#272—22no—
50C4×#272—24no—
51#6229D414yescomplemented
52C3×#550—91no—
53#5718—34no—
54C2×C32×D4C2×D318yes—
55C2×C32×D4C2×D318yes—
56C2×C32×D4C2×D318yes—
57C2×C3×#67—316no—
58C2×C3×#67—316no—
59C22×C3×Dic3—316no—
60C3×#237—632no—
61C2×C3×D12—34no—
62C2×C3×C4×D3—38no—
63C2×C3×Dic6—34no—
64C32×C42C2×D314yes—
65C3×#217—316no—
66C3×(C2×D3)⋊C4—38no—
67C3×C4⋊Dic3—34no—
68C3×C4⋊Dic3—34no—
69C3×Dic3⋊C4—38no—
70C3×C4×Dic3—38no—
71C22×H(3,𝔽3)—24no—
72C22×H(3,𝔽3)—24no—
73C22×H(3,𝔽3)—24no—
74C22×H(3,𝔽3)C2411yescharacteristic
75C2×#272—22no—
76C2×#272—22no—
77C2×#272—22no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)C2×D412yes—
80C4×H(3,𝔽3)#4112yescomplemented
81#1084—22no—
82#1084—22no—
83#1084C2×D414yes—
84C2×C3×#41—92no—
85C2×#237—38no—
86C3×#127—34no—
87C3×#127—92no—
88C3×#127C3⋊D311yescharacteristic
89C3×#121—92no—
90C3×#121—94no—
91C3×C4×Q8—91no—
92C3×C4×D4—92no—
93C3×C4×D4—91no—
94#940—38no—
95#939—316no—
96#938—38no—
97D4×Dic3—38no—
98C4×D12—34no—
99C4×Dic6—34no—
100C23×C32C22×D3116yes—
101C22×C3×D3—38no—
102C32×D4—216no—
103C32×D4—216no—
104C32×D4—48no—
105C2×C32×C4C22×D314yes—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D314yes—
108C3×#67—632no—
109C3×#67—1216no—
110C2×C3×Dic3—38no—
111C2×C3×Dic3—38no—
112C2×C3×Dic3—38no—
113C2×C3×Dic3—632no—
114C3×D12—68no—
115C3×C4×D3—616no—
116C3×Dic6—316no—
117#550—91noSylow
118C2×H(3,𝔽3)—44no—
119C2×H(3,𝔽3)C22×D411yescharacteristic
120C2×H(3,𝔽3)C2×#4111yescharacteristic
121C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
122#272—42no—
123C3×#41—188no—
124C2×C3×Q8—91no—
125C2×C3×D4—38no—
126C2×C3×D4—38no—
127C2×C3×D4—38no—
128C2×C3×D4—94no—
129C2×C3×D4—94no—
130C2×C3×D4—91no—
131C2×C3×D4C2×C3⋊D312yes—
132C2×C3×D4C2×C3⋊D312yes—
133C2×C3×D4C2×C3⋊D312yes—
134C22×C3×C4—94no—
135C22×C3×C4—92no—
136C2×#67—316no—
137C2×#67—316no—
138C22×Dic3—316no—
139#237—632no—
140C2×D12—34no—
141C2×C4×D3—38no—
142C2×Dic6—34no—
143C3×C4⋊C4—92no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—91no—
146C3×C22⋊C4—94no—
147C3×C22⋊C4—92no—
148C3×C42—34no—
149C3×C42—92no—
150C3×C42C2×C3⋊D311yescharacteristic
151#217—316no—
152(C2×D3)⋊C4—38no—
153C4⋊Dic3—34no—
154C4⋊Dic3—34no—
155Dic3⋊C4—38no—
156C4×Dic3—38no—
157C22×C32—216no—
158C22×C32—216no—
159C22×C32—216no—
160C22×C32C23×D314yes—
161C2×C3×D3—68no—
162C2×C3×D3—68no—
163C2×C3×D3—68no—
164C32×C4—24no—
165C32×C4D3×D418yes—
166C32×C4#23718yescomplemented
167C3×Dic3—316no—
168C3×Dic3—68no—
169C3×Dic3—68no—
170C2×#41—92no—
171#127—92no—
172#127#27211yescharacteristic, complemented
173#121—92no—
174#121—94no—
175C4×Q8—91no—
176C4×D4—92no—
177C4×D4—91no—
178H(3,𝔽3)#55011yescharacteristic, complemented, Sylow
179C23×C3—316no—
180C23×C3—92no—
181C23×C3C22×C3⋊D314yes—
182C22×D3—38no—
183C3×Q8—94no—
184C3×D4—24no—
185C3×D4—24no—
186C3×D4—42no—
187C3×D4—616no—
188C3×D4—616no—
189C3×D4—128no—
190C3×D4—188no—
191C3×D4—182no—
192C3×D4—364no—
193C2×C3×C4—34no—
194C2×C3×C4—34no—
195C2×C3×C4—34no—
196C2×C3×C4—92no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—188no—
200C2×C3×C4—184no—
201C2×C3×C4C22×C3⋊D311yescharacteristic
202C2×C3×C4C22×C3⋊D311yescharacteristic
203C2×C3×C4C22×C3⋊D311yescharacteristic
204#67—632no—
205#67—1216no—
206C2×Dic3—38no—
207C2×Dic3—38no—
208C2×Dic3—38no—
209C2×Dic3—632no—
210D12—68no—
211C4×D3—616no—
212Dic6—316no—
213C2×C32—416no—
214C2×C32C2×D3×D414yes—
215C2×C32C2×#23714yes—
216C2×C32#100414yes—
217C3×D3—128no—
218#41—188no—
219C2×Q8—91no—
220C2×D4—94no—
221C2×D4—94no—
222C2×D4—91no—
223C2×D4C2×#27212yes—
224C2×D4C2×#27212yes—
225C2×D4C2×#27212yes—
226C22×C4—94no—
227C22×C4—92no—
228C4⋊C4—91no—
229C4⋊C4—92no—
230C4⋊C4—91no—
231C22⋊C4—94no—
232C22⋊C4—92no—
233C42—92no—
234C42C2×#27211yescharacteristic
235C22×C3—24no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—34no—
239C22×C3—616no—
240C22×C3—616no—
241C22×C3—616no—
242C22×C3—182no—
243C22×C3—182no—
244C22×C3—182no—
245C22×C3C23×C3⋊D311yescharacteristic
246C2×D3—68no—
247C2×D3—68no—
248C2×D3—68no—
249C3×C4—21no—
250C3×C4—38no—
251C3×C4—38no—
252C3×C4—64no—
253C3×C4—94no—
254C3×C4—182no—
255C3×C4—182no—
256C3×C4D4×C3⋊D312yes—
257C3×C4#383012yes—
258Dic3—316no—
259Dic3—68no—
260Dic3—68no—
261C32#571814yescomplemented
262C23—92no—
263C23C22×#27214yes—
264Q8—94no—
265D4—24no—
266D4—24no—
267D4—42no—
268D4—188no—
269D4—182no—
270D4—364no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—92no—
274C2×C4—188no—
275C2×C4—184no—
276C2×C4C22×#27211yescharacteristic
277C2×C4C22×#27211yescharacteristic
278C2×C4C22×#27211yescharacteristic
279C2×C3—34no—
280C2×C3—34no—
281C2×C3—34no—
282C2×C3—44no—
283C2×C3—1216no—
284C2×C3—362no—
285C2×C3C2×D4×C3⋊D311yescharacteristic
286C2×C3C2×#383011yescharacteristic
287C2×C3#6436111yescharacteristic
288D3—128no—
289C22—24no—
290C22—24no—
291C22—24no—
292C22—182no—
293C22—182no—
294C22—182no—
295C22C23×#27211yescharacteristic
296C4—21no—
297C4—94no—
298C4—182no—
299C4—182no—
300C4D4×#27212yes—
301C4#8874412yescomplemented
302C3—34no—
303C3#1059618911yescharacteristic
304C2—44no—
305C2—362no—
306C2C2×D4×#27211yescharacteristic
307C2C2×#8874411yescharacteristic
308C2#1172266411yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#127C2
hypercenterC3×#127C3⋊D3
hypocenterH(3,𝔽3)#550
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#127

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#88744C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #11721850C2semidirect
4. #11721849C2semidirect
5. #11721848C2semidirect
6. D4×#1084C2semidirect
7. C4×C4⋊#272C2semidirect
8. C4×#6229C2semidirect
28. C32×#127D3semidirect
51. #6229D4semidirect
80. C4×H(3,𝔽3)#41semidirect
166. C32×C4#237semidirect
172. #127#272semidirect
178. H(3,𝔽3)#550semidirect
261. C32#5718semidirect
301. C4#88744semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#887442
H(3,𝔽3)×#1272
#117218502
#117218492
#117218482
D4×#10842
C4×C4⋊#2722
C4×#62292
C3×#57183